Place beetroot, onion, salt, vinegar, sugar and thyme in medium saucepan; bring to the boil. Reduce heat; simmer, uncovered, stirring occasionally, for 20 minutes or until beetroot is tender and slightly sticky. Cool.
Meanwhile, boil, steam or microwave cauliflower until tender. Drain; cool. Thinly slice 90g (3 ounces) of the cheddar; grate remaining cheddar.
Place cauliflower and beans in a food processor, pulse until coarsely chopped (do not over process). Transfer to a large bowl. Add ¼ cup (15g) of the breadcrumbs, the grated cheddar, parsley, rind and nuts; season, stir to combine. Shape cauliflower mixture into four patties. Refrigerate for 30 minutes.
Make lemon mayonnaise. Whisk ingredients in a small bowl until combined; season to taste.
Dip patties in egg white; coat patties in remaining breadcrumbs.
Heat oil in a large frying pan over medium-high heat; cook patties for 4 minutes each side or until browned and crisp. Drain on paper towel. Immediately top with sliced cheese for cheese to melt.
Place each patty in a lettuce leaf; top with tomato and a generous spoonful of the beetroot mixture (you will only use half the mixture, see tips). Drizzle with lemon mayonnaise; top with remaining lettuce leaf.
